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St ri p approxi m at i on wi t h Bézi er pat ches i n coni cal

f orm f or desi gn and m anuf act uri ng of devel opabl e

m at eri al s

Chi h- Hsi ng Chu a , Char l i e C. L. Wang b & Chi - Rung Tsai a

a Depar t m ent of I ndust r i al Engi neer i ng and Engi neer i ng Managem ent , Nat i onal Tsi ng Hua

Uni ver si t y , Hsi nchu, Tai wan

b Depar t m ent of Mechani cal and Aut om at i on Engi neer i ng , The Chi nese Uni ver si t y of Hong

Kong , Shat i n N. T. , Hong Kong, Peopl e' s Republ i c of Chi na

Publ i shed onl i ne: 23 Feb 2011.

To ci t e t hi s art i cl e: Chi h- Hsi ng Chu , Char l i e C. L. Wang & Chi - Rung Tsai ( 2011) St r i p appr oxi m at i on wi t h Bézi er pat ches

i n coni cal f or m f or desi gn and m anuf act ur i ng of devel opabl e m at er i al s, I nt er nat i onal Jour nal of Com put er I nt egr at ed

Manuf act ur i ng, 24: 3, 269- 284, DOI : 10. 1080/0951192X. 2010. 551782

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Strip approximation with Be ´ zier patches in conical form for design and manufacturing

of developable materials

Chih-Hsing Chua*, Charlie C.L. Wangband Chi-Rung Tsaia

aDepartment of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan;

bDepartment of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin N.T., Hong Kong,

People’s Republic of China

(Received 21 May 2010; final version received 29 December 2010)

This article proposes a novel geometric modelling method for strip approximation using developable Be ´ zier patches

in the conical form. Given two space curves, a greedy algorithm is applied to generate an aggregate of quadrilateral

and triangular patches based on local optimisation of normal variation across the patch boundaries. These patches,

only with positional continuity, are degree elevated to produce additional control points for further shape

adjustment. G1 continuity is thus created by the extra degrees of freedom obtained from the degree elevation while

the surface developability is preserved. An adaptive refinement algorithm is introduced to minimise the deviation

between the boundaries of the approximation patches and the given curves. The experimental results of test

examples demonstrate the advantages of the proposed method, which provides flexible modelling capability with

quadrilateral and triangular patches, approximates with high-order patches and allows error control in the

approximation. This work offers an effective approach for design and manufacturing of developable materials.

Keywords: developable materials; Be ´ zier patches; strip approximation; computer-aided geometric design

Introduction

Developable surfaces are a subset of ruled surfaces that

can be unfolded (or developed) into a plane without

tearing or stretching during the process. This property,

referred to as the developability, greatly eases manu-

facture of 3D objects. Therefore, geometric design with

developable (or nearly developable) shapes are used in

many industrial applications such as sheet-metal

forming (Mancewicz and Frey 1992), ship building

(Norlan 1971, Lamb 1995, Yoon et al. 2008), wind-

shield design and fabrication of apparels including

shoes and clothing (Wang et al. 2005, Cader et al. 2006,

Liu et al. 2010, Wang and Tang 2010). Parts are first

modelled with developable surfaces in 3D space. They

are then flattened into a planar pattern. The manu-

facturing process starts with cutting a material

according to the pattern. Rolling the cut material

simply resumes its original 3D shape. A final step is

often conducted to assemble different pieces by

welding or sewing in order to form the final product.

Two different approaches have been proposed for

the design of freeform developable surfaces (Chu et al.

2008). First, a surface can be represented as a tensor

product of degree (1, n) with non-linear constraints

imposed by the developability. Designers are only

allowed to specify the shape in a limited manner, e.g.

some but not all of the control points and the

remaining parameters must be solved from the con-

strained system. Previous studies employed different

techniques to simplify the solution process (Aumann

1991, Lang and Ro ¨ schel 1992, Maekawa and Chalfant

1998, Chu and Se ´ quin 2002, Aumann 2003, Chu and

Chen 2007). However, this approach may lack

practicality in modelling of complex shapes due to

the restricted degrees of freedom in the surface design

(Aumann 2004). Alternatively, one can treat a devel-

opable surface as an envelope of one parameter set of

tangent planes. The surface thus becomes a curve in

dual projective space (Pottmann and Wallner 2001).

Design methods were proposed for Be ´ zier and B-spline

surfaces based on the duality theory by Bodduluri and

Ravani 1994, Bodduluri and Ravani 1995.

Many engineering products consist of double-

curved surfaces. They are certainly not developable

in general. Practically speaking, it is tolerable to allow

deviations in the developability to a certain degree.

Materials are commonly stretched to fit design shapes

in the apparel and footwear industries. A modelling

method of truly developable surface is less important

to such products. Computer-Aided Geometric Design

(CAGD) methods that approximate 3D shape using

developable surfaces have been introduced to address

*Corresponding author. Email: chchu@ie.nthu.edu.tw

International Journal of Computer Integrated Manufacturing

Vol. 24, No. 3, March 2011, 269–284

ISSN 0951-192X print/ISSN 1362-3052 online

? 2011 Taylor & Francis

DOI: 10.1080/0951192X.2010.551782

http://www.informaworld.com

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this need. This work also falls into this category.

Several studies (Leopoldseder and Pottmann 1998,

Pottmann and Randrup 1998, Randrup 1998, Chen

et al. 1999, Pottmann and Wallner 1999) were focused

on interpolation and approximation algorithms based

on the dual approach. Leopoldseder and Pottmann

(1998) modelled a given developable surface by surfaces

of revolution. Each pair of consecutive rulings and

tangent planes that approximate the given surface is

interpolated by smoothly linked circular cones. This

work was extended to allow input as a point cloud for

applications in reverse engineering (Pottmann and

Randrup 1998, Chen et al. 1999). These methods

utilised non-linear optimisation, which makes the

computation process time-consuming when a complex

surface is modelled. On the other hand, several studies

(Frey 2002, Wang et al. 2004, Wang and Tang 2004,

Tang and Wang 2005, Wang and Tang 2005) were

focused on increasing or optimising the developability

of a surface based on its tessellation representation.

Wang et al. (2004) presented a method based on

function optimisation for increasing the developability

of a trimmed Non-uniform Rational Basis Spline

(NURBS) surface by adjusting the positions and

weights of the surface control points. The optimisation

process reduces the deformation of the original patch

while preserves G0 continuity across the boundaries of

the given trimmed surface patch. However, the

computation process is slow. Tang and Wang (2005)

proposed a modelling algorithm that interpolates two

given space curves (i.e. a strip) with an aggregate of

triangles. The interpolation task was formulated as a

variant of boundary triangulations (Frey 2002). Their

later work (Wang and Tang 2005) optimised the result

based on various objective functions originating from

different Computer-Aided

Manufacturing (CAD/CAM) applications. The solu-

tion space is limited by the fixed sampling points on the

directrices of a strip. Their global optimisation scheme

becomes inefficient when the sampling point number is

large. Shape approximation using developable patches

has also been applied to other industrial applications

such as sheet materials fabrication (Yoon et al. 2008),

tool path in five-axis flank milling (Chu and Chen 2006)

and simulation of robot motions (Lee 2004).

This article describes a geometric design method for

strip approximation using more generalised developable

surfaces. The problem is formulated as: given a strip

specified by two boundary curves in space, a set of

developable Be ´ zier patches is to be constructed to

approximate the strip bounded with a user specified

error. The method generates consecutive developable

Be ´ zier patches in the conical form, consisting of

triangular and quadrilateral patches. In the first phase,

a greedy algorithm is applied to construct an aggregate

Design/Computer-Aided

of quadratic developable patches only with positional

continuity. The next phase is to perform degree elevation

on each patch to produce extra control points for further

shape refinement. These additional degrees of freeform

(DOFs) allow us to generate G1 continuity across the

patch boundaries while maintaining the surface devel-

opability of each patch. The degree-elevated patches are

then adaptively subdivided with an optimisation scheme,

so that the approximation error between their bound-

aries and the strip can be controlled. Various test

examples illustrate the effectiveness of the proposed

algorithms. Our design method overcomes several

limitations of the past studies. It shows a more flexible

modelling capability by using quadrilateral and trian-

gular patches simultaneously. Use of quadratic patches

instead of triangles requires fewer approximating

elements and thus leads to lower problem complexity

in the following optimisation. The adaptive refinement

provides an effective mechanism of error control for the

strip approximation. This work offers a practical method

for design and fabrication of developable materials.

Preliminary

Given two curves P(u) and Q(u) in 3D space, a ruled

surface is constructed by linking each pair of corre-

sponding curve points (with equal u) with a line

segment PQ, referred to as a ruling. The surface R is

described as:

Rðt;uÞ ¼ ð1 ? tÞPðuÞþtQðuÞ;ðt;uÞ 2 ½0;1? ? ½0;1? ð1Þ

where t is the line parameter along the ruling.

Generally speaking, the tangent lines to the curves

P(u) and Q(u) at any given point do not lie in the same

plane. If these tangent lines and the corresponding

ruling remain coplanar, then the surface becomes

developable. This condition can be represented as the

triple scalar product of the two tangent vectors and the

ruling vector (P(u)7Q(u)) (Chu and Se ´ quin 2002).

(Chu and Se ´ quin 2002)

P

?ðuÞ ? Q

?

ðuÞ ? ½PðuÞ ? QðuÞ? ¼ 0

ð2Þ

Substituting the Be ´ zier representation of both curves

into Equation (2) leads to a system of equations that

must be imposed on the Be ´ zier control points to ensure

the surface developability. Previous work (Chu and

Se ´ quin 2002) derived the developability constraints from

the De Casteljau’s algorithm for simplification of the

solution process. These constraints can be characterised

with the geometry factors shown below (see Figure 1).

Geometry factor 1: For a degree-n Be ´ zier-ruled

patch, both the first two pairs and the last two pairs of

control points need to be co-planar if it is developable.

270C.-H. Chu et al.

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This property is referred to as the co-planarity

condition. However, the other control point pairs do

not have to lie in the same plane.

Geometry factor 2: For a degree-n Be ´ zier-ruled

patch satisfying the co-planarity condition, if the

extensions of all the trapezoids in the control

polyhedron converges to a single point O, the patch

becomes developable when

PiQi

PiO¼ fði ¼ 0;1;???;nÞð3Þ

where f is ascalingconstant. The Be ´ zier patch satisfying

the above condition is named as the generalised conical

form (Chu and Se ´ quin 2002), which is a subset of the

most generalised Be ´ zier developable surface.

Geometry factor 3: For a Be ´ zier-ruled patch in the

generalised conical form, we must have PiPiþ1//Qi

Qiþ1. This property can be derived from the trigono-

metric relationship specified by Equation (3).

When all pairs of PiQiare parallel to each other, i.e.

O!?, the resulting patch becomes the generalised

cylindricalform(ChuandSe ´ quin2002). WhenallPi!O,

i.e. a Be ´ zier-ruled patch with one boundary degenerated

into one single point, it is referred to as a triangular

developable patch (proof can be found in Chu and

Se ´ quin 2002). Therefore, the design of a developable

patch in the generalised conical form is to select a set of

control points Qi(or Pi) and the constant f.

Approximation with developable Be ´ zier patches:

topology construction

This section introduces a greedy algorithm to approx-

imate a given strip specified by two boundary curves

P(u) and Q(u) using an aggregate of developable Be ´ zier

patches. Since the basic constructing element can be

either a quadrilateral or a triangular patch, the goal is to

determine the topology structure of the approximating

patches from P(u) and Q(u). Note that the algorithm is

designed to prefer use of quadrilateral patches due to

the reason that quadrilateral patches may produce

fewer patch boundaries and consequently better results

in surface evaluation (see later sections). The algorithm

consists of four steps as shown in Figure 2.

Step 1: Sampling boundary curve

The two given curves are sampled into two sets of

points pis and qjs with a user specified arc length, where

the number of sampling points on P and Q does not

have to be the same.

Step 2: Feasibility test for constructing a develop-

able patch

Given a ruling piqj(see Figure 2a), there are at most

four possible ways to construct a Be ´ zier developable

patch that locally approximates the strip – their

feasibility will be tested during this step.

(1) Quadrilateral patch formed by pi7piþ17qj: we

calculate the intersection between the plane

pi7piþ17qj and the boundary curve Q (see

Figure 2b). The nearest intersection point q0jto

qjis chosen to form a candidate quadrilateral

patch pi7piþ17qj7q0j* (see Figure 2c). If

there is no intersection or the distance between

qjand q0jis greater than a distance controlled

by the user, then a feasible quadrilateral patch

does not exist.

(2) Quadrilateral patch formed by pi7qj7qjþ1:

similar to the first case, we calculate the

intersection between the curve P and the plane

pi7qj7qjþ1. If the intersection point p0iexists

and the distance between piand p0iis not greater

than user-specified value, we can construct a

quadrilateral patch with pi7qj7qjþ17 p0i**

(see Figure 2d and e); otherwise, the quad-

rilateral patch cannot be produced in this way.

(3) Triangular patch formed by pi7piþ17qj or

pi7qj7qjþ1: candidate patches can be always

produced in this way (see Figure 2f and g).

The detail of each patch construction process will

be given in the next section.

Step 3: Choose an optimal patch among candidate

patches

Step 2 may generate more than one feasible patch start-

ingwitharulingpiqj.Amongthem,theonethatgivesthe

minimal normal variation across the ruling is chosen.

The patch generation then proceeds to the next ruling.

Step 4: Repeat steps 2–3 until the end ruling is

reached

Figure 1.

in the generalised conical form.

Developability condition of a Be ´ zier-ruled patch

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Generation of a triangular patch

As described above, a triangular developable Be ´ zier

patch is defined with a projection point and a Be ´ zier

boundary curve. To construct a developable patch that

approximates the strip locally with pi7piþ17qj, we

need to consider the tangent vectors, tiand tiþ1to the

original curve at piand piþ1, respectively. As shown in

Figure 3, qjis selected as the projection point, and pi

Figure 2. Generation of candidate developable patches starting from a ruling.

272 C.-H. Chu et al.

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and piþ1are chosen to be the end control points of a

quadratic Be ´ zier curve. In some special case, an

intersection point p0i can be readily determined by

the two tangent lines defined by ti and tiþ1. More

generally, the middle point over the curve segment

pipiþ1(denoted as cpi), piand piþ1will determine a

plane S. After projecting the two curve tangent vectors

onto S (denoted as t0i and t0iþ1), we employ their

intersection p0ias the second control point. If cpi, pi

and piþ1are collinear, S is simply chosen as the plane

perpendicular to the triangle pi7piþ17qj.

Generation of a quadrilateral patch

The extensions the four control points generated by

Step 2 in the algorithm described above intersect at a

projection point O, as shown in Figure 4. The control

polygon of one boundary curve must be a scaled copy

of the other in the conical form according to Geometric

factor 2. This property indicates that only three control

points among pi, qj, piþ1and qjþ1can be freely specified

as soon as the projection point has been chosen.

Hence, the end ruling may intersect the boundary at a

different point qjþ10rather than qjþ1, given that pi, qj

and piþ1as the fixed control points. The second control

point mpiof the longer curve segment is calculated

from piand piþ1with the same heuristic adopted in

the generation of a triangular patch. Once it has

been calculated, the remaining control point mqj

of the shortersegment is

Equation (3).

thendeterminedby

Evaluation of optimal patch composition

Among all the feasible patches generated from Step 3

in the construction algorithm, the one giving the best

surface evaluation will be chosen, since it contains

two neighbouring patches with a better quality, at

least locally. This article employs normal variation as

the major surface assessment criterion. Figure 5

illustrates two consecutive patches Sk and Skþ1

connecting along the ruling piqj. Any two pairs of

the control points must be co-planar, as both

patches are in the conical form. The normal vectors

of Skand Skþ1at the ruling are denoted as nkand nkþ1,

which can be easily computed from the control

polyhedron. The normal variation across piqjis then

expressed as

NvðSk;Skþ1Þ ¼ 1 ? nk? nkþ 1

ð4Þ

* piþ1or q’jmay need to be adjusted along the ruling to

fulfil Geometric Factor 2.

**qiþ1or p’jmay need to be adjusted along the ruling

to fulfil Geometric Factor 2.

It gives the minimal value when the two normal

vectors become parallel.

Figure 3.Generation of the control polyhedron for a triangular developable Be ´ zier patch.

Figure 4.

quadrilateral developable Be ´ zier patch.

Generation of the control polyhedron for a

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Continuity adjustment with degree elevation

The consecutive patches generated based on the above

algorithm only preserve positional continuity across

the patch boundaries. However, it is advantageous to

have G1 continuity in industrial applications for

functional or aesthetic requirement, such as ship hull

design and multi-axis Computer Numerical Control

(CNC) machining. This section will introduce a

technique to create G1 continuity in the approximation

result. The main idea is to obtain additional degrees-

of-freedom for shape refinement of a Be ´ zier patch

through degree elevation. The previous work (Aumann

2004) proposed the same idea for geometric design of a

developable Be ´ zier patch. Through degree elevation,

we convert quadratic developable Be ´ zier patches into

cubic developable Be ´ zier patches, which remain the

generalised conical form. Additional control points are

thus obtained for local shape adjustment.

Degree elevation

Degree elevation is a technique widely used for

combining two curves with different degrees. For a

quadratic Be ´ zier-ruled surface with A0-A1-A2and B0-

B1-B2 as the control points of its boundary curves

(see Figure 6), the new control points for the same

surface of degree three become (Farin 1997):

A0

0¼ A0

A0

0; A0

3A0þ2

1¼1

3¼ A2; B0

3A1; A0

0¼ B0; B3¼ B2

2¼2

2¼2

1¼1

3A1þ1

3B1þ1

3A2;

B0

3B0þ2

3B1; B0

3B2

ð5Þ

The degree elevation of a triangular patch is similar

to that of a quadrilateral one but with all the control

points on one side coincident. A cubic patch elevated

from a quadratic developable patch still preserves the

developability of the surface, since degree elevation

does not change its shape. For a patch in the conical

form, the projection point remains the same and each

control point pair satisfies the scaling relationship

specified by Equation (3). On the other hand, there are

more design handles in the new representation of the

surface that can be utilised to modify the control

points with finer shape change. They allow us to

generate G1 continuity across the patch boundaries.

Continuity adjustment for quadrilateral patch

Figure 7 shows the control points of a quadrilateral

patch involved in the adjustment process. Initially, the

patch Siconnects to Si71and Siþ1along the rulings

A0B0 and A3B3, respectively, both with positional

continuity. For Si71, the tangent vectors to the

boundary curves at A0and B0are tAand tB, which

have been determined when processing the patch Si71

(i.e. they are fixed and collinear). When Siis a starting

patch, we simply assign the average of tAand tBas the

strip tangent. To achieve G1 across A0B0, we must

impose the following conditions: (1) A1 lies in the

direction of tAand (2) B1lies in the direction of tB.

To achieve G1 across A3B3is more complicated.

Instead of the original control polygon, a scaled copy

(denoted as A00-A01-A02-A03) with the last control point

A30located on the boundary is of concern. We have

A2A3//A20A30in the conical form according to Geo-

metric factor 3. Thus, it is required that (1) A2(or A20)

lies in the direction of t0Aand (2) B2lies in the direction

of t0B. To attain proximity to Si, t0Aand t0Bare chosen

as the average of the original tangent vectors on the

Figure 5.

consecutive patches.

Definition of normal variation across two

Figure 6.

surface.

Degree elevation for a quadratic Be ´ zier-ruled

274C.-H. Chu et al.

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boundary curves at A30and B3. The remaining control

points arerepresented

A2¼ A37 w2t0A. They also specify the positions of

B1and B2in a similar manner. There are two DOFs

(i.e. w1and w2) for further adjustment of the shape of

Siafter preserving the G1 continuity and the develop-

ability. The next patch Siþ1starts with a tangent vector

as t0A (or t0B). Note that all tangent vectors are

normalised in the above expressions.

asA1¼ A0þ w1tA

and

Continuity adjustment for triangular patch

When processing a triangular patch Si, the situation

becomes more complex than the quadrilateral patch –

we need to analyse the possible configurations on the

next patch Siþ1. As shown in Figure 8, if only for the

developability on the triangular patch Si, the tangent

vectors tA, tB, t0Aand t0Bdo not have to be coplanar.

However, we need to consider the constraints imposed

by the next patch Siþ1when placing these vectors in

space to achieve continuity. In detail,

(1) When Siþ1is a quadrilateral patch, we must

have t0A//t0Bso that Siþ1becomes developable.

Meanwhile, we need to have tB//t0Bso that the

G1 continuity is preserved at B0. Therefore, in

this configuration, the tangent vectors on the

end ruling must satisfy t0B//tBand t0A//t0B.

(2) When Siþ1 is a triangular patch with the

projection point at B0(or A3), we let tB//t0Bso

that G1 continuity is produced at B0. Then, t0A

is assigned to follow the boundary curve

tangent at A3.

After determining the tangent directions on the

start and end rulings of Si, the control points A1

and A2 can be expressed as A1¼ A0þ w1tA and

A2¼ A37 w2t0A. In this case, we still have two DOFs

for adjusting the shape of Siafter preserving G1 and

the surface developability.

Shape approximation

To fully determine the control polyhedron of a Be ´ zier

patch that has been degree elevated, we introduce an

error measurement into the shape approximation. The

values of w1and w2can be thus calculated based on an

optimisation scheme of the error. Suppose we use the

triangular patch Siin Figure 8 as an example. At some

proper values of w1and w2, the lower boundary curve

specified by A0-A1-A2-A3is able to approximate the

strip curve segment between A0 and A3 to a close

degree. The proximity is estimated by summation of

the deviation between the patch boundary and m

sampling points ppi (i ¼ 1, ..., m) on the strip

boundary. Therefore, w1 and w2 are computed by

solving the minimisation problem:

min

w1;w2

X

i

d2

i

ð6Þ

where diis the distance from ppito the curve specified

by A0-A1-A2-A3.

Estimating the approximation error on a quad-

rilateral patch is more complicated due to the fact that

one of the end control points may not lie on the strip

boundary (see Figure 4). Equation (6) cannot be used

because the deviation between the patch boundary

containing this control point and the strip boundary

does not necessarily indicate their real proximity. For

such a patch boundary (assume A0-A1-A2-A3), we

generate a number of rulings li(i ¼ 1, ..., m) from it

Figure 7.Continuity adjustment for a quadrilateral patch.

Figure 8.Continuity adjustment for a triangular patch.

International Journal of Computer Integrated Manufacturing275

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and treat these rulings as infinite lines. Sampling points

of the same number ppi(i ¼ 1, ..., m) are taken from

the strip curve to be compared. The approximation

error e1 is defined as summation of the shortest

distance between each ruling liand the corresponding

point ppi, as shown in Figure 9. The error measure

defined by Equation (6) still holds for the other patch

boundary. By properly adjusting the values of w1and

w2, a quadrilateral patch can approximate to the strip

boundary to a certain degree. Therefore, w1and w2

can be determined by solving the minimisation

problem:

X

min

w1;w2

i

ðd2

iþ D2

iÞð7Þ

where di is the distance from ppi to the curve

specified by B0-B1-B2-B3 and Di is the distance

from ppito lispecified by the other curve A0-A1-A2-

A3. Note that Equations (6) and (7) can be solved by

Newton’s method (Mortenson 1997, Nocedal and

Wright 1999).

Figure 9. Error estimation on a quadrilateral patch.

Figure 10.Triangular Be ´ zier developable patch refinement.

Figure 11.Quadrilateral Be ´ zier developable patch refinement.

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Adaptive refinement

The approximation error of the composite Be ´ zier

patches can be further reduced by an adaptive

refinement scheme. The main idea is to subdivide a

patch into two patches when its deviation from the

strip is greater than a user specified tolerance. One

subdivision produces two more DOFs for finer shape

control, which permit proper placement of the control

points for each subdivided patch, so that the result

becomes closer to the strip (no matter it is a triangular

or a quadrilateral patch).

For a triangular patch B0-A0-A1-A2-A3shown in

Figure 10 (the control polyhedron is drawn in thicker

lines), it will be subdivided into two triangular patches

B0-A0-A11-A12-A13and B0-A13-A21-A22-A3, where A13

Figure 12.Interpolation of a fixed boundary and a translating boundary.

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Page 11

is located at the middle point of the curve segment

between A0 and A3 on the strip boundary. Both

patches remain in the conical form. To preserve G1

continuity across the rulings B0A0and B0A3, A11must

lie on the line A0-A1and A22must lie on the line A2-A3.

The tangent direction at A13can be freely specified and

Figure 13.Interpolation of a fixed boundary and a rotating boundary.

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thus provides a design handle for shape adjustment. It

is chosen as the tangent at the middle point of the strip

boundary. A12and A21need to be located along this

direction. Each interior control point (A11, A12, A21

and A22) can still move along the tangent directions

specified above. In other words, each subdivided patch

has two free parameters to be utilised. They are

determined by minimising the error measure described

in the last section.

For a quadrilateral patch A0-A1-A2-A3-B3-B2-B1-B0

shown in Figure 11, without loss of generality, we

assume that the curve segment between A0-A3 is

shorter than the curve segment between B0-B3. Note

that the end control point A3does not necessarily lie

on the strip boundary. In patch subdivision, the new

control point A13is chosen as the middle point on the

curve segment A0-A30of the strip boundary. The

tangent on the boundary at A13is then calculated and

denoted as tm. B13must be determined by Geometric

Factor 2, maintaining the same length ratio f to A0B0

and A3B3. This indicates that the projection point O

remains unchanged during the subdivision process.

Moreover, the tangent of the subdivided curve at B13

has to be tm in order to preserve the surface

developability. The control point A12 can be thus

expressed as A12¼ A13 7 w1tm. Assuming O coin-

cident with the origin, we have B12¼ A12/f, i.e. its

location has been determined by A12. Next, A11can be

represented as A11¼ A0þ w2t0, where t0is the average

between the tangent vectors to the strip boundary at A0

and B0. Similarly, B11is defined by A11/f.

The positions of A11, A12, B11 and B12 can be

determined by minimising the approximation error

with two free parameters (i.e. w1and w2) as mentioned

in the last section. The subdivision process continues

until the error is smaller than the user specified value,

Figure 14. Interpolation of a trimmed developable patch.

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Figure 15. The result adjusted for G1 continuity and adaptively refined with different error tolerances.

280C.-H. Chu et al.

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Page 14

Figure 16.Test results of a curve pair taken from real shoe design data.

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Page 15

or the maximum number of recursions is reached. The

similar calculation procedure is applied to the other

subdivided patch defined by A13-A21-A22-A3-B13-B21-

B22-B3.

Experimental results

This section presents a number of implementation

resultsto demonstrate

approach. Input parameters for strip approximation

include: the numbers of sample point on the strip

boundaries (Np/Nq), the Hausdorff distance for

triangular patch (H3), the Hausdorff distance for

quadrilateral patch (H4) and the maximal length ratio

between the boundaries of quadrilateral patch (ra).

Figure 12 lists a series of approximation results for two

given curves (p and q). Each time one curve (p) is

translated at some distance while the other curve (q)

remains fixed. The input parameters are shown in the

figure. The numbers of produced quadrilateral patches

and triangular patches (denoted as n4 and n3)

illustrates the composition of the approximation sur-

face. Quadrilateral patches occur in the middle region,

and triangular patches are located at the ends. Fewer

quadrilateral patches are produced as the moving

distance increases. They also become more slanted.

Figure 13 shows the test results for the same curves but

this time the curve p is rotated with an angle of 158

about its shape centroid. The input parameters remain

the same as the previous example. The twist between

the curves becomes more distinct as the rotation

continues. Under this circumstance, it is less possible

to utilise quadrilateral patch in the approximation. The

results confirm that the number of quadrilateral

patches decreases with the increasing twist. The

purpose of this example is to show the capability of

the proposed method in constructing highly twisted

surfaces. The main reason why this figure may appear

odd is because the number of patches in the surface

approximation is extremely low. Most methods using

triangular patches only would fail in this condition,

while using quadrilateral and triangular patches at the

same time performs better.

Figure 14 shows the test result for one portion of a

developable surface with boundaries shown in thicker

lines. The approximation surface mainly consists of

quadrilateral patches. In fact, there is only one trian-

gular patch produced at the end (see Figure 14b). The

extension part of the quadrilateral patches can be

trimmed as shown in Figure 14c. Figure 14d displays

the test result based on the same input parameters

except with more sample points on the boundaries. The

composition of the approximation surface remains

similar, i.e. all the patches are quadrilateral except the

last one is triangular.

the effectivenessof our

Figure 15a illustrates the approximation result of

the 4th set of curves in Figure 14. The curves to be

interpolated are shown in a lighter colour. Figure

15b shows the surface that has been degree elevated

and adjusted for G1 continuity. It is intentional to

use very fewsampling

(Np¼ 4 and Nq¼ 5). As a result, the deviation of

the patch boundaries after adjustment from the strip

is large. This situation is particularly obvious in

Patch A and B. Adaptive refinement is thus applied

to improve this situation. The results corresponding

to different error tolerances are also shown. It can be

seen that the new boundaries have been ‘pulled’

towards the curves through the minimisation scheme

described in the pervious section. Moreover, more

subdivisions are required to achieve small error

control. Figure 16aillustrates

complex curves (marked as p and q) taken from a

point cloud of real shoe data. They make roughly

one quarter of the shoe surface. Figures 16b shows

the approximation result that has been adjusted for

G1 continuity,adaptively

trimmed.Figure 16cshows

unfolded from the result.

points inthis example

a pair ofmore

refined

the

andfinally

pattern planar

Conclusion

Geometric design with developable surfaces finds

many applications in product design and manufactur-

ing. Previous modelling methods for the approxima-

tion of a strip defined by two space curves have several

limitations. Only simple approximating elements like

triangles or quadrangles could be used. The approx-

imation result with these elements does not offer the

capability of shape refinement often required by real

applications. Most approaches involve complex opti-

misation processes, which become less efficient for

large numbers of approximating elements.

To overcome these problems, a novel design

method is proposed for strip approximation with

developable Be ´ zier patches in the conical form. A

greedy algorithm based on simple rules of minimising

the normal variation across the patch boundary is first

applied for generating an aggregate of patches that

interpolate two given curves. Both quadrilateral and

triangular developable patches are simultaneously used

in the process, thus offering a highly flexible capability

of strip interpolation. Quadrilateral patches would be

chosen when the tangents of the boundaries do not

deviate much from each other while triangular patches

connect the regions with more serious twist. Adoption

of polynomial patches in the approximation is

advantageous. Degree elevation is performed to create

extra degrees of freedom from the patches produced

from the first step that are only with positional

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